This book was created by 5th grade students in the Advanced Math Program at Latin American School of Monterrey, Mexico.  It contains Topics 8 to 15 of the curriculum.

This Is What We Have Learned 2

Books created by Elementary students (2nd to 6th graders) in the Advanced Math Program at Latin American School in Monterrey, Mexico.

BOOKS BY AMP STUDENTS

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A ray is a part of a line. A ray has one endpoint and goes on forever in one direction.

Congruent line segments are line segments that have the same length.

A line is a straight path of points that goes on forever in two directions.

A line segment is a part of a line with two endpoints.

Basic geometrical ideas include&nbsp;points, lines, angles, and shapes. Points are the simplest form of geometry and are represented by a dot. A line is a straight path connecting two points. Angles are formed by two lines intersecting and are measured in degrees.

1- Draw a diagram to illustrate each situation.

Pamela walks 1 mile and runs 4 miles during her daily workout.

What is the ratio of miles walked to total miles in each of Pamela’s workouts?

The absolute value of an integer is the distance from that integer to zero on the number line. Distance is always a positive measure, so the absolute value of any integer is positive.

The distance from 0 to 7 is 7 units, so |7 = 7. The distance from 0 to -7 is 7 units, so |7|=  7.

By:Mateo Villarreal, Regina Rodríguez , Alexa Rodríguez, Sofía Cantu

A Graph consist of a X coordinate  and a Y coordinate that is used to graph things like numbers

When solving equations with integers, use inverse operations to “undo” each other. Also, remember to do the same thing to both sides of an equation.

To find the possible number of combinations to distribute things you can use a table of ordered pairs

Steve has 4 pens, he wants to distribute them in 1 box and 1 pencil case. If wants to have at least 1 pen in each case or box how many possible combinations are there?

1. Make a table with boxes and cases as the labels

Subtracting an integer is the same as adding its opposite.

Rule: To subtract an integer, add its opposite.

When comparing two integers on a number line, the integer that is farther to the right is greater. The integer that is farther to the left is less.

Compare -6 and -10. Because -6 is farther to the right than -10, it is greater. So, -6 &gt; -10.

Geraldo earns $300 for hours 6 of work. If he works 8 hours at the same rate of pay, how much will he earn?

Write a proportion.     Use d for dollars earned. 

$300 $d 6 8h

Find the unit rate. Divide the first term by the second term.

Multiply by the unit rate. $50 x 8h = $400 h

So, $300 $400 6h 8h

The unit rate is $50 1 h .

Geraldo will earn $400 when he works 8 hours

By: Miranda, Ignacio, Denisse, and Ana Regina 

To complete the table we need to multiply 3x + 7 to get the value of  y. I’ll show you how.

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Graphing Equations with More Than One Operation

Use the same steps to graph an equation with more than one operation as you used to graph an equation with only one operation. Graph y = 2x - 4.

Step 1: Make a T-table. Use at least three number pairs in the table.

Step 2: Graph each ordered pair on a coordinate plane. Connect the points.

Congruent figures have the same size and shape.

Translation moves a figure in a straight direction.

The reflection of a figure gives it a mirror image over a line.

A glide reflection is a translation followed by a reflection.

 When working backwards you have to identify first what operation are you doing, if you are adding you subtract. If you are subtracting you add, and the same thing in multiply and divide

1. Identify the operation applied to the variable. 2. State the inverse operation. 3. Apply the inverse operation on both sides of the equal sign. 4. Inverse operations undo each other. 5. Solve. 6. Check by substituting.

A ratio is a comparison of two quantities. You can write a ratio in three ways:

The quantities in the ratio are called terms, so the terms in the ratio will be 3 and 4

3 to 4 3:4 or 3 4

A ratio can compare a part to a whole, a part to another part of the whole, or the whole to a part.

1:4 or 1 to 4 or 1 4

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To divide mixed numbers, you first need to change the divisor and the divident into improper fractions.

Find the reciprocal of the divisor, and then rewrite the division as a multiplication problem.

Finally, simply and solve the multiplication.

On the drawing, the scale tells us that 1 cm = 2 ft

For every 1 cm on the drawing, there are 2 ft in the kitchen.

Set up a proportion. Write the scale as the first ratio. Use the information about the kitchen for the second ratio.

A ratio table showing equal ratios can be used to solve a proportion.

Ross uses 7 pieces of fabric to make 5 pants. How many pants can he make with 35 pieces of fabric?

Make a ratio table. Multiply or divide to find equal ratios. Find ratios equivalent to 5 7 multiplying both terms of the ratio by the same number until you find 35 pieces of fabric.

So with 35 pieces of fabric you can make 25 pants

Estimating percent is the same as estimating if is 8% you estimated to 10% also if is 355% you round it ti 360%

congruent angles, have the same measure. The little symbols inside the angles, called arcs, are drawn to show congruent angles.

Vertical angles are a pair of angles formed by intersecting lights. They have no siding common vertical angles are congruent.

Adjacent angles are a pair of angles with a common vertex and a common side, but no common interior points.

When you are working with fractions and mixed numbers, you can estimate using rounding, compatible numbers, or compatible benchmark fractions.

The formula d=r x t uses symbols to relate the quantities for distance (d), average rate of speed (r), and time (t).

Distance = 15mi.              Time = 2h.                Rate = ___________

By:Mateo Villarreal, Regina Rodríguez , Alexa Rodríguez, Sofía Cantu

1 step.- Multiply the numerators and denominators. Simplify if needed. Example:

3/4 x 2/5 = (3 x 2) (4 x 5)

You can also check for cross products to simplify 

A function is a relation that a number has a specific pair and it doesn’t repeat for example

Percents Greater than 100 and less than 1

All percents can be written as fractions in simplest form and as decimals. Percents greater than 100% represent amounts greater than one whole and can be written as improper fractions and as decimals greater than 1. Percents less than 1% represent amounts less than 1 100 of the whole.

A figure has symmetry if it can be reflected to itself. It can have more than one line of symmetry.

A percent is a ratio that compares a part to a whole. The second term in the ratio is always 100. The whole is 100%.

Jana divided a sheet of paper into 5 equal sections and colored 2 of the sections red. What percent of the paper did she color?

Jana divided a sheet of paper into 5 equal sections and colored 2 of the sections red. What percent of the paper did she color?__________

Parallelogram: a four-sided quadrilateral with two pairs of equal sides.

Rectangle: A rectangle is a parallelogram with 4 right angles.

Rhombus: parallelogram with 4 equal sides

Trapezoid: quadrilateral with only 1 pair of parallel sides

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9: Division of fractions 13: Ratios

Leonardo Alibani, Diego Rivas, Rafael Mercado, Romina Vela

First make a T-table like the one at the right.

Graph each ordered pair onto the coordinate plane, then draw a line connecting the points. Every point on this line meets the condition that y = -3 + x.

Because the graph of this equation is a straight line, it is called a linear equation.

First, you need to identify the operation, and then apply inverse operations by subtracting the fraction in both sides of the equation. Finally you need to solve the equation to get the value of x.

Like addition, you need to first identify the operation, and apply inverse operations by adding the fraction on both sides of the equation. Finally, you need to solve and get the value of x.

Remember always using the steps: IDENTIFY the operation applied to the variable. • STATE the inverse operation. - addition and subtraction are inverse operations, as are multiplication and division. • APPLY the inverse operation on both sides of the equal sign. • Inverse operations UNDO each other. • SOLVE. • CHECK by substituting

How to measure an angle:

Step 1: place the protractors center on the angle’s vertex.

Step 2: place the 0° mark on one side of the angle.

Step 3: use the scale beginning with the 0° mark to read the measurement where the other side of the angle crosses the protractor.

Step 3: using the scale with the 0° mark, place a point at 52°.

Step 2: place the protractor’s center on the endpoint. Line up the ray with the 0° mark.

The order of the numbers in a ratio is important. 4:3 is the ratio of squares to circles. 3:4 is the ratio of circles to squares.

Count to find the ratio of squares to circles.

You can use this picture to find different ratios like:

The number of squares to triangles:     4 to 1, 4:1, 4/1

You can use cross multiplication to solve proportions. Gena is making costumes for the school play. She knows that 4 costumes require 20 yards of fabric. How many yards of fabric will she need for 10 costumes?

Step 1 Write a proportion. Use y for the number of yards you do not know.

4 10 20 y

4 costumes 10 costumes 20 yards y yards

4 10 20 y

y = 50 Gena will need 50 yard of fabric to for 10 costumes

Find 3+(-4). Start at 0. Move 3 units to the right. Then move 4 units to the left.

The product of two positive integers is positive. 

The product of two negative integers is positive.

The product of one positive integer and one negative integer is negative. 

Fractions, decimals, and percents all name parts of a whole. Percent means per hundred, so 15% means 15 parts per hundred.

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A ratio can compare a part to a whole, a part to another part of the whole, or the whole to a part

5 to 7 5:7 or 5/7 Arithmetic

Two equal ratios form a proportion. The units must be the same in both ratios

Write three ratios that are equal to the ratio given.

Similar figures have the same shape, but may or may not have the same size. You can use the properties of similar figures to find unknown side lengths and unknown measures of angles: • Corresponding sides in similar figures are proportional. • Corresponding angles in similar figures are congruent.

Triangles EGH and PQS are similar. What is the length of PS ? Write a proportion. EG corresponds to PQ. EH corresponds to PS. Let x= PS now solve the proportion 10/8 = 15/x 10x=120 x=12 PS= 12cm

Use unit rate to compare to rates with the same units.

Example: Two printers print paper at different rates. Printer one prints 17 papers every 3 minutes, and printer two prints 5 papers every minute. Who prints faster?

5.6 is greater than 5, so printer 1 prints faster than printer 2.

When your finding a percentage like 25% of 57, you have to transform the percent to a decimal 0.25. Then you multiply your decimal by the number your using, 0.25 times 57 = 14.25. Now we have 25% of 57.

When you’re solving a problem you always need to know if your answer is reasonable. How do you know if your answer is reasonable? Well you have to ask to yourself “ Did I use the correct process”, “ Did I answer correctly?” ” What am I asked to find? “ Does questions are what you have to ask yourself to know if your answer is reasonable.

To find a pattern, you first need to ask a question.

The questions you need to ask are: Are the numbers increasing? Are they decreasing? What is the relationship between the numbers?

In a pattern, there is always a connection between the numbers.

A rate is a ratio in which the two terms are measured in different units.

A fraction bar means division, so you can divide to find a unit rate.

In a unit rate, the second number is ALWAYS 1

Example: 27 bones for 4 dogs or 27 bones:4 dogs

The unit rate is 6.75 bones per dog. (Per means “for each”.)

The quotient of two integers with the same sign is positive.

The quotient of two integers with different signs is negative.

By: Eugenia Lozano, Elisa Miranda, Santiago Salazar

When your calculating tips you are finding the percent of a number let’s say 15% of 50. First you have to find 15% of 50 which is 7.5 so now our tip is 7.5.

When your calculating taxes you are first going to transform your percent into a decimal then multiply it by the number like 15% of 50= 7.5 so then you add it so you have to pay 57.5

To find the discount of something we have to turn the percentage to a decimal and multiply it by the number like, 15% of 50 = 7.5, now we subtract 50-7.5=52.5

To calculate simple intrest first you have to know the percent the bank is giving you let’s say 4% per year now let’s say that you deposited $1000 and left it in the bank for 4 years, we calculate this by 1000x0.4x4 =160 so that’s how much you earned in 4 years

Triangles are figures with only 3 sides, a triangle is 1/2 of a square. There are 3 types of triangles equilateral, a triangle with all equal sides; isosceles, a triangle with 2 equal sides; and a scalene, a triangle with no equal sides.

There are also the angles in each triangle. In a equilateral triangle is angle measures 60º; there are also right triangles( a triangle that has a right angle, an acute triangle ( are angles are less than 90º ), and an obtuse angle ( one angle measure more than 90º).

Multiplying a Fraction and a Whole Number

Find 3/5 of 15, or 3/5 x 15. 15 divided by 5 = 3, so 1/5 x 15 = 3 Since 3/5 is 3 times 1/5, 3/5 x 15 = 3 x ( 1/5 x 15) = 3 x 3 =9 3/5 x 15 = 9

Examples: 4/5 x 20 = 16 6/7 x 14 = 12

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